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Chapter 7: Problem 32

Expand each expression. $$ (3 k-2 m)^{2} $$

Short Answer

Expert verified
Expanding (3k - 2m)^2 gives 9k^2 - 12km + 4m^2.

Step by step solution

01

Identify the Binomial to Expand

Consider the binomial (3k - 2m)^2 This is in the form of (a - b)^2 , where a = 3k and b = 2m.
02

Apply the Binomial Formula

The formula for expanding (a - b)^2 is a^2 - 2ab + b^2 . Substitute a = 3k and b = 2m into the formula.
03

Substitute and Simplify

Substitute the values of a and b into a^2 - 2ab + b^2 to get: (3k)^2 - 2(3k)(2m) + (2m)^2.Let's simplify each term separately.
04

Simplify Each Term

(3k)^2 = 9k^2 -2(3k)(2m) = -12km (2m)^2 = 4m^2
05

Combine the Terms

Putting it all together, (3k - 2m)^2 = 9k^2 - 12km + 4m^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Expressions
In algebra, expanding expressions is an essential skill. It involves transforming a mathematical expression into a more detailed and spread-out form. For example, consider the binomial expression \(3k - 2m\)^2. This is a simpler way of writing a more expanded form. By expanding, we turn it into a polynomial with multiple terms.
The process of expanding is crucial for simplifying expressions and solving equations more efficiently. In this example, we apply algebraic rules to multiply the binomial and make it easier to work with.
Binomial Formula
The binomial formula is a powerful tool in algebra that helps in expanding expressions involving binomials. A binomial is simply an expression with two terms, like \(a - b\). The binomial formula provides a way to square or raise such expressions to higher powers without having to multiply them out manually each time.
For instance, the formula for \(a - b\)^2 is \(a^2 - 2ab + b^2\). This saves time and reduces the chance of errors. When working with our example, \(3k - 2m\)^2, we identify \a = 3k\ and \b = 2m\, and apply the binomial formula to get the expanded expression.
Algebraic Manipulation
Algebraic manipulation involves using various algebraic rules and operations to transform expressions. This includes expanding, factoring, simplifying, and rearranging terms. In the case of \(3k - 2m\)^2, algebraic manipulation helps us break down the expression and simplify it step by step.
Here鈥檚 a brief rundown: First, we identify our values of \a\ and \b\. Then using the binomial formula, we perform the following steps:
  • Square the first term: \(3k\)^2 = 9k^2.
  • Multiply the two terms by two: \-2(3k)(2m) = -12km.
  • Square the second term: \(2m\)^2 = 4m^2.
Finally, we combine all the terms to get the expanded expression: \9k^2 - 12km + 4m^2\.
Polynomials
Polynomials are expressions that consist of variables and coefficients, structured in terms like \ax^n\, where \ is a non-negative integer. Expanding a binomial like \(3k - 2m\)^2 results in a polynomial. This polynomial includes terms with different degrees.
Our expanded expression from \(3k - 2m\)^2 is \9k^2 - 12km + 4m^2\. Here, \9k^2\ is a term with degree 2\, \-12km\ is a term with degree 1 in both variables, and \4m^2\ is a term with degree 2 in \m\.
Understanding polynomials helps you see how various algebraic expressions come together. It is a foundation for learning more advanced topics in algebra and beyond.

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